Meanwhile, heat the oil in a large skillet over medium-high heat. Add the sausage and cook, breaking it up with a spoon, until browned, 6 to 7 minutes.

Add the Brussels sprouts, 1/2 teaspoon salt, and 1/4 teaspoon pepper and cook, tossing, for 2 minutes. Add the chicken broth and bring to a simmer. Serve over the ravioli and sprinkle with cheese, if desired.